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f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 | import OAI.Probability.FreeIsing.Brownian.MeanSquareDescent
import OAI.Probability.FreeIsing.Ising.CanonicalCoordinateIdentification
import OAI.Probability.FreeIsing.Ising.CanonicalCovarianceIntegrals
import OAI.Probability.FreeIsing.Ising.ObservationPosteriorMartingale
import OAI.Probability.FreeIsing.Ising.PosteriorIntegralRepresentation
import OAI.Probability.FreeIsing.Martingales.IntegerIdentification
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology
namespace Problem367.IsingObservationFactor
open IsingObservationInnovation
variable {d : ℕ} {beta : ℝ}
variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
theorem canonical_root_covariance_integrals_hasSum
(hd : 3 ≤ d) (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1)
(T : ℝ≥0) (J : TreeVertex d → Paths d → ℝ)
(hJ : ∀ w, MemLp (J w) 2 (observationLaw μ))
(hJlim : ∀ w, Tendsto (fun n => ∫ x,
(BrownianGrid.uniformSum (fun s x => innovationMap beta x w s)
(continuousRootCovariance hb w) T n x - J w x) ^ 2 ∂observationLaw μ)
atTop (𝓝 0)) :
HasSum (fun w => (hJ w).toLp (J w))
((posterior_memLp beta μ (ConnectedHull.root d) T 2).toLp
(posterior beta (ConnectedHull.root d) T)) := by
let B : ℝ≥0 → ℝ → ℝ := fun s r => WienerSampler.sample r s
have hmB (s : ℝ≥0) : Measurable (B s) :=
(continuous_eval_const s).measurable.comp WienerSampler.measurable_sample
have hcB (r : ℝ) : Continuous (fun s => B s r) :=
(WienerSampler.sample r).continuous
obtain ⟨X, I, hI, hM, hX, hXlim, hXI, _hmI, _htI, _hzero, hsum, _herror⟩ :=
IsingInnovation.exists_posterior_integral_representation hμ hd hb hcrit
uniform01 WienerSampler.sample_isBrownianReal.toIsPreBrownianReal
hmB hcB (ConnectedHull.root d) T
have hIJ (w : TreeVertex d) : I w =ᵐ[μ.prod (labelMeasure d)]
J w ∘ uniformObservation := by
exact canonical_covariance_coordinate_integral_eq_pullback hd hμ hb hcrit w T
(fun n => hX n w) (hI w) (fun n => hXlim n w) (hXI w) (hJ w) (hJlim w)
exact PredictableL2.hasSum_toLp_descend_of_comp_ae
(measurePreserving_uniformObservation μ) J hJ
(posterior_memLp beta μ (ConnectedHull.root d) T 2)
I hI hM hIJ (Eventually.of_forall (fun _ => rfl)) hsum
theorem canonical_root_covariance_integrals_hasSum_all_time
(hd : 3 ≤ d) (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1)
(J : TreeVertex d → ℝ≥0 → Paths d → ℝ)
(hJM : ∀ w, Martingale (J w) (filtration d) (observationLaw μ))
(hJ2 : ∀ w t, MemLp (J w t) 2 (observationLaw μ))
(hJlim : ∀ w (N : ℕ), 0 < N → ∀ t ≤ (N : ℝ≥0),
Tendsto (fun n => ∫ x,
(BrownianIntegralProcess.stoppedUniformSum
(fun s x => innovationMap beta x w s)
(continuousRootCovariance hb w) N n t x - J w t x) ^ 2
∂observationLaw μ) atTop (𝓝 0)) (t : ℝ≥0) :
HasSum (fun w => (hJ2 w t).toLp (J w t))
((posterior_memLp beta μ (ConnectedHull.root d) t 2).toLp
(posterior beta (ConnectedHull.root d) t)) := by
apply MartingaleIntegerIdentification.hasSum_of_positive_integer hJM
(posterior_martingale hμ (ConnectedHull.root d)) hJ2
(fun s => posterior_memLp beta μ (ConnectedHull.root d) s 2) ?_ t
intro N hN
have happrox (w : TreeVertex d) : Tendsto (fun n => ∫ x,
(BrownianGrid.uniformSum (fun s x => innovationMap beta x w s)
(continuousRootCovariance hb w) N n x - J w N x) ^ 2 ∂observationLaw μ)
atTop (𝓝 0) := by
simpa only [BrownianIntegralProcess.stoppedUniformSum_terminal] using
hJlim w N hN N le_rfl
have hterminal := canonical_root_covariance_integrals_hasSum hd hμ hb hcrit
(N : ℝ≥0) (fun w => J w N) (fun w => hJ2 w N) happrox
exact PredictableL2.tendsto_integral_sq_error_of_hasSum
(fun w => J w N) (fun w => hJ2 w N)
(posterior_memLp beta μ (ConnectedHull.root d) N 2) hterminal
theorem exists_canonical_root_covariance_integral_representation
(hd : 3 ≤ d) (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) :
∃ (J : TreeVertex d → ℝ≥0 → Paths d → ℝ)
(hJ2 : ∀ w t, MemLp (J w t) 2 (observationLaw μ)),
(∀ w, IsStronglyPredictable (filtration d) (J w)) ∧
(∀ w, Martingale (J w) (filtration d) (observationLaw μ)) ∧
(∀ w, ∀ᵐ x ∂observationLaw μ, Continuous (fun t => J w t x)) ∧
(∀ w, J w 0 =ᵐ[observationLaw μ] 0) ∧
(∀ w t, (∫ x, J w t x ∂observationLaw μ) = 0) ∧
(∀ w t, (∫ x, J w t x ^ 2 ∂observationLaw μ) =
∫ s in (0 : ℝ)..(t : ℝ), ∫ x,
continuousRootCovariance hb w s.toNNReal x ^ 2 ∂observationLaw μ) ∧
(∀ w (N : ℕ), 0 < N → ∀ t ≤ (N : ℝ≥0),
Tendsto (fun n => ∫ x,
(BrownianIntegralProcess.stoppedUniformSum
(fun s x => innovationMap beta x w s)
(continuousRootCovariance hb w) N n t x - J w t x) ^ 2
∂observationLaw μ) atTop (𝓝 0)) ∧
(∀ t, HasSum (fun w => (hJ2 w t).toLp (J w t))
((posterior_memLp beta μ (ConnectedHull.root d) t 2).toLp
(posterior beta (ConnectedHull.root d) t))) := by
obtain ⟨J, hJp, hJM, hJc, hJ2, hJ0, hJmean, hJenergy, hJlim⟩ :=
exists_canonical_root_covariance_integrals hd hμ hb hcrit
exact ⟨J, hJ2, hJp, hJM, hJc, hJ0, hJmean, hJenergy, hJlim,
canonical_root_covariance_integrals_hasSum_all_time hd hμ hb hcrit J hJM hJ2 hJlim⟩
end Problem367.IsingObservationFactor
end
end OAI
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